The Terms in Lucas Sequences Divisible by Their Indices

Chris Smyth
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
James Clerk Maxwell Building
King's Buildings
Mayfield Road
Edinburgh EH9 3JZ
United Kingdom

Abstract:

For Lucas sequences of the first kind
and second kind
defined as usual by
,
, where and are either integers or
conjugate quadratic integers, we describe the sets
divides and
divides .
Building on earlier work, particularly that of Somer, we show that the
numbers in these sets can be written as a product of a so-called basic number, which can only be , or , and particular
primes, which are described explicitly. Some properties of the set of
all primes that arise in this way is also given, for each kind of
sequence.